16-bit binary grid

J, 26 bytes

('   ';' # '){~4 4$_16{.#:

An anonymous verb. Thankfully, J is very good at drawing boxes. Let's try it out:

   f =. ('   ';' # '){~4 4$_16{.#:
   f 4242
+---+---+---+---+
|   |   |   | # |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+
| # |   |   | # |
+---+---+---+---+
|   |   | # |   |
+---+---+---+---+

As some commenters have mentioned, the way J draws boxes is system-dependent: on some platforms, this code will work under the default settings, but on others, the boxes will be drawn using Unicode line drawing characters. (The commands 9!:6 and 9!:7 allow you to query and set the characters to draw boxed values with, respectively.)

JavaScript (ES6), 102

... or 96 using return instead of console.log.

Test running the snippet below in an EcmaScript 6 compliant browser.

f=n=>{for(o=h=`
+---+---+---+---+
`,z=16;z--;n/=2)o=(z&3?'':h+'|')+` ${' #'[n&1]} |`+o;console.log(o)}

// TEST
console.log=x=>O.innerHTML=x+O.innerHTML

function test(n) { f(n); console.log(n); }

<input id=I value=4680><button onclick='test(+I.value)'>-></button>
<pre id=O></pre>

Befunge-93, 196 218 bytes

&00p12*v>>4>"---+",v v <
v*:*:*:<   | :-1,,,< #
>:*2/10p^  >"+",25*,10g|
     > #v^#         $< @
 25*,^  >4" |",,v ,*<>
v>"#",00g10g-00p 10g
 |`-1g01g00     <>48^
v>" ",10g
>2/10p>"| ",,1-:#^_$>^

To run the program...

1. Go to the online interpreter.
2. Paste this code in the big text box.
3. Click Show.
4. Input the desired number in the Input box.
5. Click Run. (Or change Slow to something like 5 milliseconds and then click Show.)
6. Ta-da!

Output for 4242:

+---+---+---+---+
|   |   |   | # |
+---+---+---+---+
|   |   |   |   |
+---+---+---+---+
| # |   |   | # |
+---+---+---+---+
|   |   | # |   |
+---+---+---+---+

Output for 33825:

+---+---+---+---+
| # |   |   |   |
+---+---+---+---+
|   | # |   |   |
+---+---+---+---+
|   |   | # |   |
+---+---+---+---+
|   |   |   | # |
+---+---+---+---+

Explanation

Oh goodness, what have I got myself into? Well, here goes! (Irrelevant code is replaced with .s.)

Part 1: Get input (store in 0,0) and calculate 32768 (store in 1,0).

&00p12*v>
v*:*:*:< 
>:*2/10p^

Part 2: Print out "+---+---+---+---".

>4>"---+",v
  | :-1,,,<

Part 3: Print "+" and a newline and check to see if (1,0) is 0 (i.e. we're done). If so, terminate. Otherwise, continue.

 ........... v <
   | ....... # 
   >"+",25*,10g|
v.#         $< @
>4" |",,v ...

Part 4: Get binary digits of input, updating (0,0) and (1,0) as we go along. Print the right things. I take advantage of Befunge's wrap-around behavior.

 .....  >4" |",,v ,*<.
v>"#",00g10g-00p 10g
 |`-1g01g00     <>48^
v>" ",10g
>2/10p>"| ",,1-:#^_...

Part 5: Print a newline and go back to the part that prints "+---+---+---+---+". Wrap-around trick is used.

     > #.^.         .. .
 25*,^  ......... ...>
................ ...
 .........      .....
........
.................._$>^

Ta-da!